Theorem singoutnupword

Description: Singleton with character out of range S is not an increasing sequence for that range. (Contributed by Ender Ting, 22-Nov-2024)

	Ref	Expression
Hypothesis	singoutnupword.1	$$⊢A\in \mathrm{V}$$
Assertion	singoutnupword	$$⊢\neg A\in S\to \neg ⟨“A”⟩\in UpWordS$$

Proof

Step	Hyp	Ref	Expression
1		singoutnupword.1	$$⊢A\in \mathrm{V}$$
2	1	singoutnword	$$⊢\neg A\in S\to \neg ⟨“A”⟩\in \mathrm{Word}S$$
3		upwordisword	$$⊢⟨“A”⟩\in UpWordS\to ⟨“A”⟩\in \mathrm{Word}S$$
4	2 3	nsyl	$$⊢\neg A\in S\to \neg ⟨“A”⟩\in UpWordS$$